Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials (2009.01418v2)

Abstract: $N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $\beta$-Hermite, $\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $\beta\to\infty$ in the $\beta$-Hermite and $\beta$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $\Sigma_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $\Sigma_N^{-1}$ with formulas for the eigenvalues and eigenvectors of $\Sigma_N^{-1}$ and thus of $\Sigma_N$. In the present paper we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for $\Sigma_N$ from $\Sigma_N^{-1}$ where, for $\beta$-Hermite and $\beta$-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for $N\to\infty$ in terms of the Airy function. For $\beta$-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.